MM 2016 Exam 1 EF Questions PDF

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

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Student Name:

EXAM FACTOR ABN: 15568488769 www.examfactor.com.au

MATHEMATICAL METHODS Units 3 & 4 Trial examination 1 2016 Reading time: 15 minutes Writing time: 1 hour

QUESTION AND ANSWER BOOK Structure of book Number of questions

Number of questions to be answered

Number of marks

10

10

40

• Students are permitted to bring into the examination room, pens, pencil, highlighters, erasers, sharpeners, rulers. • Students are not permitted to bring into the examination room: notes of any kind, blank sheets of paper, white out liquid/tape or a calculator of any type. Materials supplied • Questions and answer book of 12 pages, with a detachable sheet of miscellaneous formulas. • Working space is provided throughout the book. Instructions • Detach the formula sheet. • Write your name in the space provided above on this page. • All written responses must be in English.

Students are NOT permitted to bring mobile phones and/or any other unauthorized electronic devices into the examination room.

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Answer all questions in the spaces provided. In all questions where a numerical answer is required an exact value must be given unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale.

Question 1 (4 marks)

x2 − 5 , find f ′( x) . e2 x

a.

If f ( x ) =

b.

If f x = sin 2 3x , find f ′ π .

()

( )

2 marks

( )

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2 marks

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 2 (5 marks) a.

Find the anti–derivative of

b.

Given that

1

( 2x − 3)

2

with respect to x.

2 marks

4

∫ f (x) dx = 6 , find the value of 0 4

∫ ( f (x) + 2) dx

2 marks

0

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 3 (4 marks) If f (x) = 1− x and g(x) = e x

(

)

a.

Show that g f (x) exists.

b.

Find g f (x) and give its domain.

c.

Evaluate

(

2 marks

)

g ( f (0 ) ) .

1 mark

1 mark

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 4 (3 marks) Solve the equation f ( x) = e x − 3e −x = 2, for x.

Question 5 (4 marks)

(

)

( )

Solve the equation 2 log2 x − 1 = −2 + log2 2x for x.

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 6 (4 marks)

π⎞ ⎛ For the function y = 3sin 2 ⎜ x − ⎟ , for −π ≤ x ≤ π 4⎠ ⎝ a.

b.

Give the amplitude and period.

2 marks

π⎞ ⎛ Sketch the graph of y = 3sin 2⎜ x − ⎟ , for − π ≤ x ≤ π on the set of axes below. 4⎠ ⎝ Show all relevant intercepts with the axes and the coordinates of the endpoints. 2 marks

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 7 (4 marks) There is a daily bus from Manangatang to Melbourne but bad weather can cause delays. The probability of the bus departing on time, given good weather is 0.8 and the probability of the bus leaving on time, given bad weather is 0.6. In November the probability that on a particular day the weather is good is 0.7. Find the probability that on a particular day in November a.

the bus from Manangatang to Melbourne departs on time.

2 marks

b.

the weather is good, given the bus departs on time.

2 marks

TURN OVER

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 8 (3 marks) Let X be a random variable with a normal distribution with a mean of 8 and a standard deviation of 2. Let Z be a random variable with the standard normal distribution and Pr Z < −2 = 0.02 .

(

)

a.

Find Pr (X > 12).

b.

Find Pr( X < 8 | X < 12) . Express your answer as a fraction.

1 mark

2 marks

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 9 (5 marks) A random sample of n, undergraduate university students were asked whether they were going to complete a second degree. Sixty percent percent said no. a.

What is the value of the sample proportion, pˆ .

b.

Write an expression for e, the margin of error for this estimate at the 95% confidence level in terms of the sample size, n. 2 marks

c.

If the number of people in the sample were halved what would be the effect on e. 2 marks

1 mark

TURN OVER

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Question 10 (5 marks) The function f (x) = ax 3 , x ≥ 0 , has a tangent at point A as shown on the diagram.

a.

Find the coordinates of A in terms of a.

The average value of the function, f, between the origin and the point A is b.

Determine the value of a.

2 marks

2 . 8

3 marks

END OF QUESTION AND ANSWER BOOK

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Mathematical Methods Formulas Mensuration 1 (a + b)h 2

volume of a pyramid:

1 Ah 3

curved surface area of a cylinder:

2π rh

volume of a sphere:

4 π r3 3

volume of a cylinder:

π r 2h

area of a triangle:

1 bcsin A 2

volume of a cone:

1 π r 2h 3

area of a trapezium:

Calculus d n −1 x = nxn dx

∫ x dx = n + 1 x

d ax e = aeax dx

∫e

d n n−1 ( ax + b) = an (ax + b ) dx

∫( ax + b)

d (log e ( x) ) = 1 dx x

∫ x dx = log (x) + c, x > 0

d (sin(ax)) = a cos( ax) dx

∫ sin(ax )dx = − a cos(ax) + c

d (cos( ax)) = − a sin( ax) dx

∫ cos (ax) dx = a sin(ax) + c

( )

( )

(

1

n

)

ax

n+1

+ c, n ≠ −1

1 dx = e ax + c a n

dx =

1 (ax + b)n+1 + c,n ≠ −1 a(n + 1)

1

e

1

1

d (tan( ax)) = 2a = a sec2 (ax ) dx cos (ax ) product rule:

du dv d (uv) = u +v dx dx dx

quotient rule:

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d ⎛u⎞ ⎜ ⎟= dx ⎝ v ⎠

v

dv du −u dx dx 2 v

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2016 MATHEMATICAL METHODS 3 & 4 EXAM 1

Probability Pr( A) = 1 − Pr( A' ) Pr(A B ) =

Pr( A ∪ B) = Pr( A) + Pr( B) −Pr( A ∩ B)

Pr( A ∩ B) Pr( B)

µ = E( X )

mean

variance

Probability Distribution discrete

continuous

Pr( X = x) = p(x) Pr( a < X < b) =

∫

Mean

Variance

µ = Σ x p(x)

σ 2 = Σ ( x − µ ) 2 p( x)

b

f ( x) dx

var(X ) = σ 2 = E((X − µ ) 2 ) = E (X 2 ) − µ 2

µ=

∫

∞

∞

x f ( x) dx

−∞

−∞

a

σ 2 = ∫ ( x − µ) 2 f ( x) dx

Sample proportions X Pˆ = n standard deviation

ˆ = sd(P)

p(1 − p ) n

mean

ˆ =p E( P)

approximate confidence interval

ˆ ⎞ ⎛ ˆ − pˆ ) p(1 pˆ (1 − p) ˆ , pˆ + z ⎜⎝ p − z n n ⎟⎠

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